CCN COMPLEX COMPUTING NETWORKS1 This research has been supported in part by European Commission FP6 IYTE-Wireless Project (Contract No: )

CCN COMPLEX COMPUTING NETWORKS 2 The System Modelling and Circuit Implementation from Time-Frequency Domain Signal Specifications By F. Acar Savacı

CCN COMPLEX COMPUTING NETWORKS3 Overview of The Proposed System Signal Synthesis System Modeling Circuit Synthesis TimeFrequency Time x(t) Time V out Time Frequency Verification (w,D,b,c,b)

CCN COMPLEX COMPUTING NETWORKS4 Signal Synthesis

CCN COMPLEX COMPUTING NETWORKS5 Wavelet Transform The wavelet transform is used to represent the signals in time- frequency domain by using a basis including the scaled and translated versions of the mother wavelet. The advantage of this transform among the other time- frequency domain transforms is the multiresolution property. The slowly changing features and the high frequency components of a signal can be observed simultaneously with the multiresolution property.

CCN COMPLEX COMPUTING NETWORKS6 Wavelet Transform The continuous wavelet transform of s (t) Admissibility condition

CCN COMPLEX COMPUTING NETWORKS7 Advantages of Wavelet Transform

CCN COMPLEX COMPUTING NETWORKS8 Energy in Time-Frequency Domain The local time-frequency energy density normalized scalogram: The total energy of the signal Discrete Wavelets

CCN COMPLEX COMPUTING NETWORKS9 Wavelet Ridges Wavelet ridges are the curves in time-frequency plane where the energy is localized. The wavelet method follows the rapid variations of the instantaneous frequencies (IF) by adapting the width of the window according to the frequency. The instantaneous frequency of a signal is simply defined as the derivative of the phase of the analytical signal and can be found by determining the wavelet ridges of the signal.

CCN COMPLEX COMPUTING NETWORKS10 By the stationary phase theorem, the wavelet transform isthe stationary phase theorem When the mother wavelet has a spectral peak at,the wavelet transform is maximized at the stationary points by neglecting the effect of the correction term. These points are called as the ridge of the wavelet transform given as:

CCN COMPLEX COMPUTING NETWORKS11 Wavelet Ridges of a Hyperbolic Chirp

CCN COMPLEX COMPUTING NETWORKS12 Wavelet Ridges for Multi-Component Signals The wavelet transform of a multi-component signal Therefore, the wavelet transform coefficients are localized along the L curves

CCN COMPLEX COMPUTING NETWORKS13 Reconstruction of Signals from Wavelet Ridges Approximate signal:

CCN COMPLEX COMPUTING NETWORKS14 Signal Synthesis System Modeling Circuit Synthesis TimeFrequency Time x(t) Time V out Time Frequency Verification (w,D,b,c,b)

CCN COMPLEX COMPUTING NETWORKS15 System Modelling The wavelet network which combines feedforward neural networks and wavelet decompositions using a learning algorithm of backpropagation type has been proposed for nonlinear static and dynamical system modelling. For static systems:

CCN COMPLEX COMPUTING NETWORKS16 Static Wavelet Network

CCN COMPLEX COMPUTING NETWORKS17 Dynamical System Modelling Assume that a process is generating Let a vector is constructed by Taken’s embedding theorem asembedding follows the dynamical evolution of the original system. Therefore, the next state of the system is predicted from the past observations.

CCN COMPLEX COMPUTING NETWORKS18

CCN COMPLEX COMPUTING NETWORKS19 Signal Synthesis System Modeling Circuit Synthesis TimeFrequency Time x(t) Time V out Time Frequency Verification (w,D,b,c,b)

CCN COMPLEX COMPUTING NETWORKS20 Circuit Synthesis Mexican Hat Mother Wavelet The function has been implemented with antilog amplifier for exponential function, adders, multipliers and amplifers.

CCN COMPLEX COMPUTING NETWORKS21 The Circuit Implementation of Dynamical Wavelet Network

CCN COMPLEX COMPUTING NETWORKS22 Signal Synthesis System Modeling Circuit Synthesis TimeFrequency Time x(t) Time V out Time Frequency Verification (w,D,b,c,b)

CCN COMPLEX COMPUTING NETWORKS23 Verification Singular Value Decomposition Based Ridge Determination Singular Value Decomposition Based Ridge Determination The Carmona methods work well for noisy signals but their computational cost is high especially for the long signals with spread spectrum. The energy in the wavelet domain is described by the scalogram matrix. With singular value decomposition, the dominant and less significant components of a matrix can be decomposed. Therefore, the singular value decomposition of the scalogram matrix have been used to smooth the noise effect in the scalogram. And then the ridges can be found using simple method.

CCN COMPLEX COMPUTING NETWORKS24 Applications: Monocomponent Signal Signal Synthesis Desired Specifications Obtained Signal WT of Obtained Signal

CCN COMPLEX COMPUTING NETWORKS25 Applications: Monocomponent Signal System Modelling Embedded Signal in Phase Space Output of Dynamical Wavelet Network in MATLAB

CCN COMPLEX COMPUTING NETWORKS26 Applications: Monocomponent Signal Circuit Synthesis Output of Dynamical Wavelet Network Circuit in SPICE

CCN COMPLEX COMPUTING NETWORKS27 Applications: Monocomponent Signal-Verification Output of Dynamical Wavelet Network Modulus of WT of the Output of Dynamical Wavelet Network Phase of WT of the Output of Dynamical Wavelet Network Instant. Frequency of the Output of Dynamical Wavelet Network

CCN COMPLEX COMPUTING NETWORKS28 Applications: Multicomponent Aperiodic Signal Signal Synthesis Desired Specifications Obtained Signal WT of Obtained Signal

CCN COMPLEX COMPUTING NETWORKS29 Applications: Multicomponent Aperiodic Signal System Modelling Embedded Signal in Phase Space Output of Dynamical Wavelet Network in MATLAB

CCN COMPLEX COMPUTING NETWORKS30 Applications: Multicomponent Aperiodic Signal Circuit Synthesis Output of Dynamical Wavelet Network Circuit in SPICE

CCN COMPLEX COMPUTING NETWORKS31 Applications: Multicomponent Aperiodic Signal-Verification Output of Dynamical Wavelet Network Modulus of WT of the Output of Dynamical Wavelet Network Phase of WT of the Output of Dynamical Wavelet Network Instant. Frequency of the Output of Dynamical Wavelet Network

CCN COMPLEX COMPUTING NETWORKS32 Conclusion The nonlinear circuit with an output signal which has the desired time-frequency domain energy localizations can be synthesized by using the proposed method Although the number of the components of the circuit may be large compared to the nonlinear dynamical systems exhibiting several nonlinear phenomena, it is suitable for VLSI manufacturing because of the systematic procedure.

CCN COMPLEX COMPUTING NETWORKS33 Conclusion Since the ridges obtained from nonstationary signals can give the relevant information about the phase structures of the dynamical systems, the time-frequency domain may be the suitable domain for giving the specifications of the output of the nonlinear dynamical circuit. The SVD-based ridge determination algorithm can be used for noise reduction since the singular value decomposition of the scalogram matrix can be successfully used for the extraction of the dominant energy components.

CCN COMPLEX COMPUTING NETWORKS34 Conclusion The wavelet network is a powerful tool for modelling the nonlinear dynamical systems. However, the selection of the mother wavelet plays a crucial role in the approximation. The proposed method may be used in music synthesis applications to implement the musical compositions or in any sound synthesis applications. By observing the wavelet transform characteristics of the signals from nature, the approximate characteristics can be implemented with the given method.

CCN COMPLEX COMPUTING NETWORKS35 Discrete Transform Discrete case a 0 : Dilation step size b 0 : Translation step size The total energy

CCN COMPLEX COMPUTING NETWORKS36 Scalogram Matrix Finite number of discrete dilation and translation coefficients Back

CCN COMPLEX COMPUTING NETWORKS37 Stationary Phase Theorem Back

CCN COMPLEX COMPUTING NETWORKS38 Selection of Embedding Delay The time delay should be some multiple of the sampling time since only these observations are present, long enough to evolve the system short enough not to loose the correlation between the samples Usually, first minimum of average mutual information is chosen

CCN COMPLEX COMPUTING NETWORKS39 Selection of Embedding Dimension False nearest neighborhood method Distance in d-dimensional space is compared with the distance in d+1dimensional space If the additional distance is large compared to then the points in the phase space are the false neighbors. The dimension is increased until the geometric structure of the attractor is unfolded. Back

CCN COMPLEX COMPUTING NETWORKS40 Singular Value Decomposition : Diagonal matrix of singular values (orthogonal matrices )

CCN COMPLEX COMPUTING NETWORKS41 Energy in Wavelet Domain Related with The Singular Values

CCN COMPLEX COMPUTING NETWORKS42 Approximated Scalogram Matrix When the noise is additive white Gaussian noise, the smaller singular values are more affected. Therefore, truncation of the scalogram reduces effect of the noise. The noise-free approximated scalogram For the chosen noise threshold ,

CCN COMPLEX COMPUTING NETWORKS43 Ridges The points in the wavelet domain such that at the fixed time instant the maximum energy occurs at the scale The ridges with the sufficiently large amount of the energy are also included in the reconstruction of the signal and these ridges are defined as Back